75 research outputs found
On a Dynamical Brauer-Manin Obstruction
Let F : X --> X be a morphism of a variety defined over a number field K, let
V be a K-subvariety of X, and let O_F(P)= {F^n(P) :n=0,1,2,...} be the orbit of
a point P in X(K). We describe a local-global principle for the intersection of
V and O_F(P). This principle may be viewed as a dynamical analog of the
Brauer-Manin obstruction. We show that the rational points of V(K) are
Brauer--Manin unobstructed for power maps on P^2 in two cases: (1) V is a
translate of a torus. (2) V is a line and P has a preperiodic coordinate. A key
tool in the proofs is the classical Bang-Zsigmondy theorem on primitive
divisors in sequences. We also prove analogous local-global results for
dynamical systems associated to endomoprhisms of abelian varieties.Comment: 17 page
On the reduction of a non-torsion point of a Drinfeld module
AbstractLet Ï be a Drinfeld Fq[T]-module defined over a global function field K. Let zâK be a non-torsion point. We prove that for almost all monic elements nâFq[T] there exists a place â of K such that n is the âorderâ of the reduction of z modulo â
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